Ricci flat metrics with bidimensional null orbits and non-integrable orthogonal distribution. (English) Zbl 1123.53023

In the present article the author finds all metrics \(g\) in a \(4\)-dimensional manifold of any signature with vanishing Ricci tensor, that satisfy three conditions: (i) \(g\) allows a Lie algebra of Killing vector fields with \(2\)-dimensional orbits, (ii) the tensor \(g\) degenerates when restricted to the orbits, and (iii) the distribution orthogonal to the orbits is not integrable. The main result is that there exists only one (up to sign), Ricci flat metric.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
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